Recently, Melham computed some nite sums in which the denominator of the summand includes product... more Recently, Melham computed some nite sums in which the denominator of the summand includes products of `sine' or `cosine'. In this paper, generalizations of the sums, which he studied in 2016, are presented, by allowing arbitrary factors in the denominator of the summand. Note more than the elementary technique of partial fraction decomposition method is used. Furthermore, some of the sums, which he studied in 2017, are treated in the same style.
In this paper, we make a contribution to the enumeration of permutations avoiding a quadruples of... more In this paper, we make a contribution to the enumeration of permutations avoiding a quadruples of 4-letter patterns by establishing a Wilf class composed of 19 symmetry classes.
In this paper, we will present various results on computing of wide classes of Hessenberg matrice... more In this paper, we will present various results on computing of wide classes of Hessenberg matrices whose entries are the terms of any sequence. We present many new results on the subject as well as our results will cover and generalize earlier many results by using generating function method. Moreover, we will present a new approach on computing Hessenberg determinants, whose entries are general higher order linear recursions with arbitrary constant coefficients, based on finding an adjacency-factor matrix. We will give some interesting showcases to show how to use our new method.
We consider new kinds of max and min matrices, [ a max(i,j) ] i,j≥1 and [ a min(i,j) ] i,j≥1 , as... more We consider new kinds of max and min matrices, [ a max(i,j) ] i,j≥1 and [ a min(i,j) ] i,j≥1 , as generalizations of the classical max and min matrices. Moreover, their reciprocal analogues for a given sequence {an} have been studied. We derive their LU and Cholesky decompositions and their inverse matrices as well as the LU-decompositions of their inverses. Some interesting corollaries will be presented.
We introduce a nonsymmetric matrix defined by q-integers. Explicit formulæ are derived for its LU... more We introduce a nonsymmetric matrix defined by q-integers. Explicit formulæ are derived for its LU-decomposition, the inverse matrices L−1 and U−1 and its inverse. Nonsymmetric variants of the Filbert and Lilbert matrices come out as consequences of our results for special choices of q and parameters. The approach consists of guessing the relevant quantities and proving them later by traditional means.
In this paper, we present both a new generalization and an analogue of the Filbert matrix F by th... more In this paper, we present both a new generalization and an analogue of the Filbert matrix F by the means of the Fibonacci and Lucas numbers whose indices are in nonlinear form λ (i + r) k + µ (j + s) m + c for the positive integers λ, µ, k, m and the integers r, s, c. This will be the first example as nonlinear generalizations of the Filbert and Lilbert matrices. Furthermore we present q-versions of these matrices and their related results. We derive explicit formulae for the inverse matrix, the LU-decomposition and the inverse matrices L −1 , U −1 as well as we present the Cholesky decomposition for all matrices.
The reciprocal super Catalan matrix studied by Prodinger is further generalized, introducing two ... more The reciprocal super Catalan matrix studied by Prodinger is further generalized, introducing two additional parameters. Explicit formulae are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the qversion of Zeilberger's celebrated algorithm.
A class of symmetric band matrices of bandwidth 2r + 1 with the binomial coefficients entries was... more A class of symmetric band matrices of bandwidth 2r + 1 with the binomial coefficients entries was studied in [5]. We consider a class of non-symmetric band matrices with the Gaussian q-binomial coefficients whose upper bandwith is s and lower bandwith is r. We give explicit formulae for determinant, inverse and LU-decomposition of the class. We compute the value of infinity-norm of the inverse matrix H −1 for the case q → 1. We use the q-Zeilberger algorithm and unimodality property to prove claimed results.
The Sylvester matrix was first defined by JJ Sylvester. Some authors have studied the relationshi... more The Sylvester matrix was first defined by JJ Sylvester. Some authors have studied the relationships between certain orthogonal polynomials and the determinant of the Sylvester matrix. Chu studied a generalization of the Sylvester matrix. In this paper, we introduce its 2-periodic generalization. Then we compute its spectrum by left eigenvectors with a similarity trick.
Article 10.5.8.] computed partial in…nite sums including reciprocal usual Fibonacci, Pell and gen... more Article 10.5.8.] computed partial in…nite sums including reciprocal usual Fibonacci, Pell and generalized order-k Fibonacci numbers. In this paper we will present generalizations of earlier results by considering more generalized higher order recursive sequences with additional one coe¢ cient parameter.
We present generalizations of Ruehr’s identities with two additional parameters. We prove the cla... more We present generalizations of Ruehr’s identities with two additional parameters. We prove the claimed results by two different proof methods, namely combinatorially and mechanically. Further, we derive recurrence relations for some special cases by using the Zeilberger algorithm.
Recently, Melham computed some nite sums in which the denominator of the summand includes product... more Recently, Melham computed some nite sums in which the denominator of the summand includes products of `sine' or `cosine'. In this paper, generalizations of the sums, which he studied in 2016, are presented, by allowing arbitrary factors in the denominator of the summand. Note more than the elementary technique of partial fraction decomposition method is used. Furthermore, some of the sums, which he studied in 2017, are treated in the same style.
In this paper, we make a contribution to the enumeration of permutations avoiding a quadruples of... more In this paper, we make a contribution to the enumeration of permutations avoiding a quadruples of 4-letter patterns by establishing a Wilf class composed of 19 symmetry classes.
In this paper, we will present various results on computing of wide classes of Hessenberg matrice... more In this paper, we will present various results on computing of wide classes of Hessenberg matrices whose entries are the terms of any sequence. We present many new results on the subject as well as our results will cover and generalize earlier many results by using generating function method. Moreover, we will present a new approach on computing Hessenberg determinants, whose entries are general higher order linear recursions with arbitrary constant coefficients, based on finding an adjacency-factor matrix. We will give some interesting showcases to show how to use our new method.
We consider new kinds of max and min matrices, [ a max(i,j) ] i,j≥1 and [ a min(i,j) ] i,j≥1 , as... more We consider new kinds of max and min matrices, [ a max(i,j) ] i,j≥1 and [ a min(i,j) ] i,j≥1 , as generalizations of the classical max and min matrices. Moreover, their reciprocal analogues for a given sequence {an} have been studied. We derive their LU and Cholesky decompositions and their inverse matrices as well as the LU-decompositions of their inverses. Some interesting corollaries will be presented.
We introduce a nonsymmetric matrix defined by q-integers. Explicit formulæ are derived for its LU... more We introduce a nonsymmetric matrix defined by q-integers. Explicit formulæ are derived for its LU-decomposition, the inverse matrices L−1 and U−1 and its inverse. Nonsymmetric variants of the Filbert and Lilbert matrices come out as consequences of our results for special choices of q and parameters. The approach consists of guessing the relevant quantities and proving them later by traditional means.
In this paper, we present both a new generalization and an analogue of the Filbert matrix F by th... more In this paper, we present both a new generalization and an analogue of the Filbert matrix F by the means of the Fibonacci and Lucas numbers whose indices are in nonlinear form λ (i + r) k + µ (j + s) m + c for the positive integers λ, µ, k, m and the integers r, s, c. This will be the first example as nonlinear generalizations of the Filbert and Lilbert matrices. Furthermore we present q-versions of these matrices and their related results. We derive explicit formulae for the inverse matrix, the LU-decomposition and the inverse matrices L −1 , U −1 as well as we present the Cholesky decomposition for all matrices.
The reciprocal super Catalan matrix studied by Prodinger is further generalized, introducing two ... more The reciprocal super Catalan matrix studied by Prodinger is further generalized, introducing two additional parameters. Explicit formulae are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the qversion of Zeilberger's celebrated algorithm.
A class of symmetric band matrices of bandwidth 2r + 1 with the binomial coefficients entries was... more A class of symmetric band matrices of bandwidth 2r + 1 with the binomial coefficients entries was studied in [5]. We consider a class of non-symmetric band matrices with the Gaussian q-binomial coefficients whose upper bandwith is s and lower bandwith is r. We give explicit formulae for determinant, inverse and LU-decomposition of the class. We compute the value of infinity-norm of the inverse matrix H −1 for the case q → 1. We use the q-Zeilberger algorithm and unimodality property to prove claimed results.
The Sylvester matrix was first defined by JJ Sylvester. Some authors have studied the relationshi... more The Sylvester matrix was first defined by JJ Sylvester. Some authors have studied the relationships between certain orthogonal polynomials and the determinant of the Sylvester matrix. Chu studied a generalization of the Sylvester matrix. In this paper, we introduce its 2-periodic generalization. Then we compute its spectrum by left eigenvectors with a similarity trick.
Article 10.5.8.] computed partial in…nite sums including reciprocal usual Fibonacci, Pell and gen... more Article 10.5.8.] computed partial in…nite sums including reciprocal usual Fibonacci, Pell and generalized order-k Fibonacci numbers. In this paper we will present generalizations of earlier results by considering more generalized higher order recursive sequences with additional one coe¢ cient parameter.
We present generalizations of Ruehr’s identities with two additional parameters. We prove the cla... more We present generalizations of Ruehr’s identities with two additional parameters. We prove the claimed results by two different proof methods, namely combinatorially and mechanically. Further, we derive recurrence relations for some special cases by using the Zeilberger algorithm.
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Papers by Talha Arıkan